Poll
0%
25%
25%
25%
25%
0%
Originally posted by Barker
The name of the Greek letter π is pi, and this spelling is used in typographical contexts where the Greek letter is not available or where its usage could be problematic. When referring to this constant, the symbol π is always pronounced like "pie" in English, the conventional English pronunciation of the letter.The constant is named π because it is the first letter of the Greek words "περιφέρεια" (transliterated: periphereia; periphery in English) and "περίμετρον" (perimetron, perimeter). The Swiss mathematician Leonhard Euler proposed that this number be given a particular name and suggested the use of π.
The numerical value of π truncated to 50 decimal places is:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
See the links below and those at sequence A00796 in OEIS for more digits.
With the 50 digits given here, the circumference of any circle that would fit in the observable universe (ignoring the curvature of space) could be computed with an error less than the size of a proton.[1] Nevertheless, the exact value of π has an infinite decimal expansion: its decimal expansion never ends and does not repeat, since π is an irrational number (and indeed, a transcendental number). This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer. See history of numerical approximations of π.
The constant π is an irrational number; that is, it cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert.
Furthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.
herbnone
(Copyrighted contents from PurpleMath removed)
Originally posted by Barker
physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. Among the characteristics of chaotic systems, described below, is sensitivity to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, the behavior of systems that exhibit chaos appears to be random, even though the system is deterministic in the sense that it is well defined and contains no random parameters. Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economics, and population growth.Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder. (See the article on mythological chaos for a discussion of the origin of the word in mythology, and other uses.) A related field of physics called quantum chaos theory studies non-deterministic systems that follow the laws of quantum mechanics.
As well as being orderly in the sense of being deterministic, chaotic systems usually have well defined statistics. For example, the Lorentz system pictured is chaotic, but has a clearly defined structure. Weather is chaotic, but its statistics - climate - is not.
For a dynamical system to be classified as chaotic, most scientists will agree that it must have the following properties:
* it must be sensitive to initial conditions,
* it must be topologically mixing, and
* its periodic orbits must be dense.Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.
Sensitivity to initial conditions is popularly known as the "butterfly effect", suggesting that the flapping of a butterfly's wings over Tokyo might create tiny changes in the atmosphere, which could over time cause a tornado to occur over Texas. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.
Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the phase space of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from x to 2x). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behavior, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems.
Even for bounded systems, sensitivity to initial conditions is not identical with chaos. For example, consider the two-dimension torus described by a pair of angles (x,y), each ranging between zero and 2π. Define a mapping that takes any point (x,y) to (2x, y + a), where a is any number such that a/2π is irrational. Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial conditions. However, because of the irrational rotation in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic according to the definition above.
Topologically mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is an example of a chaotic system.
Bifurcation diagram of a logistic map, displaying chaotic behavior past a threshold
The first discoverer of chaos can plausibly be argued to be Jacques Hadamard, who in 1898 published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature. In the system studied, Hadamard's billiards, Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one-another, with positive Lyapunov exponent. In the early 1900s, Henri Poincaré while studying the three-body problem, found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Much of the early theory was developed almost entirely by mathematicians, under the name of ergodic theory. Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff, A.N. Kolmogorov, M.L. Cartwright, J.E. Littlewood, and Stephen Smale. Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.
Chaos theory progressed more rapidly after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical. One of the earliest electronic digital computers, ENIAC, was used to run simple weather forecasting models.
An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a basic computer, a Royal McBee LGP-30, to run his weather simulation. He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.
To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The printout rounded variables off to a 3-digit number, but the computer worked with 6-digit numbers. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.
Yoshisuke Ueda independently identified a chaotic phenomenon as such by using an analog computer on November 27, 1961. The chaos exhibited by an analog computer is truly a natural phenomenon, in contrast with those discovered by a digital computer. Ueda's supervising professor, Hayashi, did not believe in chaos throughout his life, and thus he prohibited Ueda from publishing his findings until 1970.
The term chaos as used in mathematics was coined by the applied mathematician James A. Yorke.
The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research.
🤨
Originally posted by PeterGriffin
:In biology, evolution is the change in the heritable traits of a population over successive generations, as determined by shifts in the allele frequencies of genes. Through the course of time, this process results in the origin of new species from existing ones (speciation). All contemporary organisms are related to each other through common descent, the products of cumulative evolutionary changes over billions of years. Evolution is the source of the vast diversity of extant and extinct life on Earth.[1][2]The basic mechanisms that produce evolutionary change are natural selection (which includes ecological, sexual, and kin selection) and genetic drift; these two mechanisms act on the genetic variation created by mutation, genetic recombination and gene flow. Natural selection is the process by which individual organisms with favorable traits are more likely to survive and reproduce. If those traits are heritable, they are passed to succeeding generations, with the result that beneficial heritable traits become more common in the next generation.[3][4][5] Given enough time, this passive process can result in varied adaptations to changing environmental conditions.[6]
The modern understanding of evolution is based on the theory of natural selection, which was first set out in a joint 1858 paper by Charles Darwin and Alfred Russel Wallace and popularized in Darwin's 1859 book The Origin of Species. In the 1930s, Darwinian natural selection was combined with the theory of Mendelian heredity to form the modern evolutionary synthesis, also known as "Neo-Darwinism". The modern synthesis describes evolution as a change in the allele frequency within a population from one generation to the next.[6]
The theory of evolution has become the central organizing principle of modern biology, relating directly to topics such as the origin of antibiotic resistance in bacteria, eusociality in insects, and the staggering biodiversity of the living world. The modern evolutionary synthesis is broadly received as scientific consensus and has replaced earlier explanations for the origin of species, including Lamarckism, and is currently the most powerful theory explaining biology.
Because of its potential implications for the origins of humankind, evolutionary theory has been at the center of many social and religious controversies since its inception.
Part of the Biology series on
Evolution
Mechanisms and processesAdaptation
Genetic drift
Gene flow
Mutation
Selection
Speciation
Research and historyEvidence
History
Modern synthesis
Social effect
Evolutionary biology fieldsEcological genetics
Evolutionary development
Human evolution
Molecular evolution
Phylogenetics
Population genetics
Evo-devo
Biology Portal · v·d·e
Contents
[hide]* 1 Study of evolution
o 1.1 History of evolutionary thought
+ 1.1.1 Classical Darwinian theory
+ 1.1.2 Modern synthesis
o 1.2 Molecular genetics
o 1.3 Academic disciplines
* 2 Evidence of evolution
o 2.1 Morphological evidence
o 2.2 Molecular evidence
o 2.3 Evidence from studies of complex iteration
* 3 Ancestry of organisms
o 3.1 History of life
* 4 Modern synthesis
o 4.1 Heredity
o 4.2 Variation
o 4.3 Mechanisms of evolution
+ 4.3.1 Mutation
+ 4.3.2 Recombination
+ 4.3.3 Gene flow and Population structure
+ 4.3.4 Drift
+ 4.3.5 Horizontal gene transfer
+ 4.3.6 Selection and adaptation
o 4.4 Speciation and extinction
* 5 Current Research
o 5.1 Micro RNA's
* 6 Misunderstandings about modern evolutionary biology
o 6.1 Distinctions between theory and fact
o 6.2 Evolution and devolution
o 6.3 Speciation
o 6.4 Self-organization and entropy
o 6.5 Information
* 7 Social and religious controversies
* 8 Notes
* 9 Additional references
* 10 External links
* 11 See also[edit]
Study of evolution
Main article: Evolutionary biology
[edit]
History of evolutionary thought
Main article: History of evolutionary thought
Charles Darwin in 1854, five years before publishing The Origin of Species.
Charles Darwin in 1854, five years before publishing The Origin of Species.The idea of biological evolution has existed since ancient times, notably among Greek philosophers such as Anaximander and Epicurus and Indian philosophers such as Patañjali. Scientific theories of evolution were proposed in the 18th and 19th centuries, by scientists such as Jean-Baptiste Lamarck and Charles Darwin.
[edit]
Originally posted by Barker
In Albert Einstein's theories of relativity time dilation is manifested in two circumstances:* In special relativity, clocks that are moving with respect to an inertial system of observation (the putatively stationary observer) are found to be running slower. This effect is described precisely by the Lorentz transformations.
* In general relativity, clocks at lower potentials in a gravitational field-- such as in close proximity to a planet --are found to be running slower. This gravitational time dilation is only briefly mentioned in this article but is described elsewhere (see also gravitational red shift).
The account of the effect that is specific to Special Relativity is as follows. As viewed from the "stationary" reference system the clock slowdown due to relative speed is a "real" change in the time properties of the object under observation. Any and all such observations made from reference systems not co-moving with the object, will confirm this effect. The effect is reciprocal: as observed from the point of view of the system we have heretofore regarded as moving, it will be the other party's clocks that have slowed. (This description presumes that the relative motion of both parties is uniform; that is, they do not accelerate with respect to one another during the course of the observations.)
In contrast, gravitational time dilation (as treated in General Relativity) is not reciprocal: an observer at the top of a tower will observe that clocks at ground level tick slower, and observers on the ground will agree. Thus gravitational time dilation is agreed upon by all stationary observers, independent of their altitude.
The formula for determining time dilation in special relativity is:
where
Δ t0 is a measured time interval attributed to a "moving" clock,
Δ t is that same time interval as measured in the "stationary" system of reference,is the Lorentz factor,
v is the relative speed between the clock and the stationary system, and
c is the speed of light.Thus the duration of the clock cycle of a moving clock appears to be increased: it is "running slow." As indicated, the Lorentz transforms can be used for more general cases.
As shown, the effect increases in an exponential manner with respect to relative speed or gravitational differences. The range of such variances in ordinary life, even considering space travel, are not great enough to produce easily detectable time dilation effects, and such vanishingly small effects can be safely ignored. It is only when an object approaches speeds on the order of 30,000 km/s (1/10 of the speed of light), or lies deep within the gravitational "well" of massive stellar objects, that it becomes important.
Time dilation by the Lorentz factor was predicted by Joseph Larmor (1897), at least for electrons orbiting a nucleus. Thus "... individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 😬sqrt{1 - v^2/c^2}" (Larmor 1897). Time dilation of magnitude corresponding to this (Lorentz) factor has been experimentally confirmed, as described below.
Interestinghmm
Originally posted by PeterGriffin
Because of the humor style of the show, snippets of Peter's history prior to the start of the series are extremely ambiguous, simply because events are often shown simply for their humor value, with no thought put into whether or not it fit into continuity (ranging from Peter loitering for an extended period of time, to Peter having received a sex change, and even several with his apparent death).Peter Griffin was born in Quahog, Rhode Island, around 1960, to Francis and Rose Griffin. He was baptized in the Roman Catholic faith as a baby, but did not grow up to be very devout.
According to Peter's doctor, Peter is a Cancer; more specifically he was born in July, meaning he was born between July 1 to July 22.
He was a member of the folk-singing group Simon & Garfunkel, where he pitched ideas for "Here's To You Mrs. Fleckenstein" and "Parsley, Sage, Rosemary, and Lawry's Seasoning Salt". When Simon & Garfunkel turned down these ideas he said "Screw you guys, I'm goin' to 'Nam".
During the Vietnam War, Peter participated on the American side. Instead of camouflage, he wore a bright clown's get-up, reasoning that "they're gonna be looking for Army guys.". Despite actually fighting in the war, he believes the Walter Cronkite hoax of the war never actually happened. (Though Peter fighting in the Vietnam War may have been a joke and not a real occurrence, as according to his age, he would only have been a child at the time.)
Around 1980, Peter worked at a country club/resort in Newport, RI, as a towel boy, where he met Lois Pewterschmidt, the daughter of a wealthy industrialist. Lois and Peter got to know each other better at a party for the resort employees that Lois sneaked off to. Peter dropped the girl he was dancing with to the floor and started dancing with Lois to "Do You Love Me?" by the Contours.
Her father, Carter, objected to Peter because he considered him to be of a lower social class, while Peter's father objected to Lois because of her religion. Carter tried to keep Lois and Peter apart by having his servants toss Peter out into the ocean. Unfortunately for Carter, Peter was rescued by a nearby Navy ship, aboard which Peter met his future neighbor, Glenn Quagmire. Glenn helped Peter pull into a port in Florida. There he met Cleveland Brown, another future neighbor, who gave him a ride back to Newport to Lois. Upon returning, Carter Pewterschmidt offered Peter a check for $1 million to not marry Lois, but Peter turned it down. When Peter and Lois got married, Francis Griffin (Peter's father) amended the "Just Married" banner to read "Just Married - to a Protestant whore."
Peter worked as a production line worker at the Happy-Go-Lucky Toy Factory since at least 1977, until the plant was torn down a couple of decades later, after owner Mr. Weed died.
When a toad-licking drug trend started at Meg's school, Peter went "undercover" as a Fonzie-inspired high school student under the name Lando Griffin (in part an homage to Lando Calrissian from Star Wars). Meg was at first mortified, but when Lando became cool by single-handedly turning the entire school off drugs, Meg asked him to the dance. Lando took Connie D'Amico to the dance instead, but declared that he had been rejected by Meg, and promised to kill himself by driving his motorcycle off a cliff. Ironically, Peter has himself tried a number of drugs, including LSD, marijuana, anabolic steroids, cocaine, ecstasy and opium.
After being laid off from the toy factory, Peter bought a boat and became a fisherman. When his boat sank during a trip to Pelican's Reef, Peter and his drinking buddies were stranded on an island for a number of months. Upon their rescue, Peter returned to Quahog to find that Lois had married Brian following his disappearance. Showing up naked at the Spooner Street house one day, he managed to win back the affections of Lois, who reached a mutual agreement to divorce Brian and remarry Peter.
Peter worked at the Pawtucket Patriot brewery for a few months, during which time he drank himself into a stupor in his first five minutes of working there. However, in an earlier episode, "Wasted Talent", the brewery was presented as a 'Willy Wonka' type factory, complete with 'Oompa Loompas' (Called 'Chumba Wumbas' in this episode). In the episode 'Wasted Talent' the brewery has restricted access, and Peter was thrown out.
Peter once played as a center in the New England Patriots as the number 68. He got chosen when he bulldozed at least eighty people trying to go to the bathroom, which Tom Brady liked. He was fired for his end-zone showboating at a game and was traded to the London Silly Nannies. Peter decided to get back at Brady, and called him to schedule a football game. As the kickoff started the whole London Silly Nannies left except for Peter who Tom Brady thought was a man to stand up to the entire New England Patriots.
In the movie, Peter became a local celebrity when he hosted an Action 5 News segment titled "What Really Grinds My Gears" (a reference to a line from Planes, Trains and Automobiles) in which he ranted on topics such as the lack of new priest and rabbi jokes, Lindsay Lohan's teasing, parents who don't control their children, not being able to find droids you're looking for (like the Imperial Stormtroopers in Star Wars), and people in the 19th century. He lost the job when Tom Tucker brought to the station a tape of Stewie driving while intoxicated.
Like Homer Simpson, Peter has had a wide variety of jobs. See List of Peter Griffin's jobs.
[edit]Personality
Peter's favorite pastime is watching television. His favorite shows include Star Trek, Three's Company, Happy Days, Joan of Arcadia, Gumbel 2 Gumbel, Who's the Boss and Lost.
Peter is insanely jealous of Lois' ex-boyfriends, and he will attack any man who expresses the slightest interest in her; he even punched an orca at Sea World after it "kissed" Lois, and his own reflection after Lois said, "Well, look at that handsome man!" Two notable exceptions to this include: upon learning that Lois had been sexually involved with Gene Simmons, a member of his favorite band, KISS, he was proud of her, even boasting, "my wife did KISS!" and that he "feels like [he] did KISS too". Also, when the Griffin's neighbor Quagmire was caught peeping at Lois, Peter sided with Quagmire and told Lois she should take peeping as a compliment and to forgive him.
Sexuality
Although heterosexual, there have been a few throwaway jokes over the course of the series implying Peter is gay. However, it's possible that the fact of Peter's retardation as stated in the episode "Petarded" that plays a role in homosexual acts.
Originally posted by Barker
A barker is a person who attempts to attract patrons to entertainment events, such as a carnival, by exhorting passing public, describing attractions of show and emphasizing variety, novelty, beauty, or some other feature believed to incite listeners to attend entertainment. A barker may conduct a brief free show, introducing performers and describing acts to be given at the feature performance.A barker is also a southern term for a prostitute in the UK.
dodgy
Originally posted by PeterGriffin
Seann William Scott Is 'Big Brother'
Friday, September 8th, 2006 - [Big Brother]
Joining the mentoring program...Quaid, Weisz, Church Join 'Smart People'
Friday, September 8th, 2006 - [Smart People]
In the indie dramedy...Meg Ryan In 'Homeland Security' Talks
Friday, September 8th, 2006 - [Homeland Security]
With Antonio Banderas...'God Grew Tired Of Us' For National Geographic
Friday, September 8th, 2006 - [God Grew Tired of Us]
Sundance Film Festival winner...Shawn Levy In Talks To Direct 'Matchbreaker'
Friday, September 8th, 2006 - [Matchbreaker]
A high-concept comedy...Baby Spice To Replace Beyonce In 'Pink Panther 2'
Thursday, September 7th, 2006 - [Pink Panther 2]
Beyonce is too busy with album...Kenneth Branagh To Direct Drama 'Sleuth'
Thursday, September 7th, 2006 - [Sleuth]
Which stars Jude Law and Michael Caine...Timothy Olyphant To Join Army 'Stop Loss'
Thursday, September 7th, 2006 - [Stop-Loss]
Joesph Gordon-Levitt also joins...Tom McCarthy To Write & Direct 'The Visitor'
Thursday, September 7th, 2006 - [The Visitor]
Revolves around a widower...Jim Carrey & Cameron Diaz Join 'A Little Game'
Wednesday, September 6th, 2006 - [A Little Game]
As an engaged Manhattan couple...Alamode Film Takes On 'Ten Canoes'
Wednesday, September 6th, 2006 - [Ten Canoes]
The Australian Oscar candidate...'Exhausting Being Fabulous' For Galan Entertainment
Wednesday, September 6th, 2006 - [It's Exhausting Being Fabulous]
Based on Richard Perez-Feria's memoirs...Snoot Entertainment To Make 'Identity Theft'
Wednesday, September 6th, 2006 - [Identity Theft]
Based on the futuristic novella...Truly Indie To Distribute '51 Birch Street'
Tuesday, September 5th, 2006 - [51 Birch Street]
Movie is directed by Amy Seplin...Warner Bros. To Delay 'Blood Diamond'
Tuesday, September 5th, 2006 - [The Blood Diamond]
Ed Zwick's Leonardo DiCaprio starrer...Charlize Theron Set For 'Battle In Seattle'
Friday, September 1st, 2006 - [Battle in Seattle]
With Stuart Townsend directing...'Bond 22' Set For Nov. 7, 2008 Release
Friday, September 1st, 2006 - [Bond 22]
Put back from May...Angela Bettis Set For 'Scar' Horror
Friday, September 1st, 2006 - [Scar]
In the psychological horror...Supermodel Gemma Ward Joins 'The Strangers'
Friday, September 1st, 2006 - [The Strangers]
Alongside Live Tyler...Emily Mortimer Joins 'Lars And The Real Girl'
Friday, September 1st, 2006 - [Lars and the Real Girl]
Opposite Ryan Gosling...Peter Jackson To Remake 'The Dam Busters'
Thursday, August 31st, 2006 - [The Dam Busters]
Remake of the 1954 war film...Harold Perrineau To Star In '28 Weeks Later'
Thursday, August 31st, 2006 - [28 Weeks Later]
Star of TV series Lost...Eva Birthistle Joins The 'Nightwatching' Cast
Thursday, August 31st, 2006 - [Nightwatching]
The Peter Greenaway's Rembrandt biopic...Intrepid Pictures To Develop Horror 'Mercy'
Thursday, August 31st, 2006 - [Mercy]
Written by Carolyn Townsend...More Spider-Man After 'Spider-Man 3'
Wednesday, August 30th, 2006 - [Spider-Man 4]
Next one not the last...